ADMISSIBLE CYCLIC REPRESENTATIONS AND AN ALGEBRAIC APPROACH TO QUANTUM PHASE

Nonadmissible, weakly admissible and admissible cyclic representations and other algebraic properties of the generalized homographic oscilla tor (GHO) are studied in detail. For certain ranges of the deformation parameter, it is shown that this new deformed oscillator is a prototy pe cyclic oscillator endowed with a non-negative (admissible) spectrum . By changing the deformation parameter, the cyclic spectrum can be tu ned to have an arbitrarily large period. It is shown that the standard harmonic oscillator is recovered at the nonadmissible infinite-period limit of the GHO. With these properties, the GHO provides a concrete example of an oscillator rich in a variety of cyclic representations. It is well known that such representations are of relevance to the pro per algebraic formulation of the quantum-phase operator. Using a gener al scheme, it is shown that admissible cyclic algebras permit a well-d efined Hermitian phase operator of which properties are studied in det ail at finite periods as well as at the infinite-period limit; Fujikaw a's index approach is applied to admissible cyclic representations and in particular to the phase operator in such algebras. Using the speci fic example of GHO it is confirmed that the infinite-period limit is d istinctively singular. The connection with the Pegg-Barnett phase form alism is established in this singular limit as the period of the cycli c representations tends to infinity. The singular behaviour at this li mit identifies the algebraic problems, in a concrete example, emerging in the formulation of a standard quantum harmonic-oscillator phase op erator.